Thank you for all the kind replies!
Just to clarify the 3-hour thing, I was referring to distances by public transport. As you all were saying, connections exist but getting from A to B seems to take several times as long by public transport as by car.
I was hoping someone would come up with a suggestion that would magically prove to be the exception to the rule. I will look up the places that have been mentioned upthread.
What I hate about driving: fearing that I miss my turn and complex situations (e.g. trying to find and get into a parking lot on a narrow street with lots of traffic around me). Also not hot on getting onto very busy motorways. I can deal with it if must, but prefer someone sitting in the front scanning whatever situation is coming up and confirming my reactions. However, if I'm travelling only with DH and DC, he'll be in the back with DC (and would be of limited help in the front anyway).
I've managed driving in a continental capital and on the motorways around it, but did find it very stressful (and mostly had a co-pilot in the front). In the UK I've only actually driven in rural Yorkshire, and kept mainly to the country lanes as I preferred avoiding driving in towns.
So - I could probably deal with getting from a spacious parking lot on a beach via country lanes to an out-of-town sight with another spacious parking lot. Driving to places in a large town, or even driving through a large town, is something I'd rather avoid if I can!
What sort of place I'd like to stay, apart from the transport and availability constraints: not glamping and not a large town. A place with natural beauty and where I feel like I'm breathing fresh air as opposed to thick car fumes (which is my daily existence in a city). Close to a beach.
Secondary desirables: easy access to food (supermarkets are fine, a good restaurant would be lovely but not essential), something of cultural interest.
I do realise that I can't have the moon on a stick...
And thank you again for all the advice!